Stress-Aligned Hexahedral Lattice Structures

With this implementation you can generate stress-aligned hexahedral lattice structures as described in the paper.

Dependencies

Used libraries can be easily installed with pip install -r requirements.txt.

Required are

• scikit-fem for all FEM related things (requires meshio)
• NumPy for vectorized arrays,
• SciPy for (sparse) matrix operations,
• libigl for fast winding numbers and
• drbutil for various utilities.

Optional and recommended are

• CuPy for CUDA accelerated NumPy,
• PyPardiso for a parallelized CPU solver,
• embreex for fast ray intersections,
• trimesh as fallback option for other methods,
• tqdm for progress bars,
• Mayavi and Polyscope for nice visuals.

Run Examples

In the main directory you can run python runExamples.py to generate example results. This script contains basic setups to recreate results (2D & 3D) from the paper and an overview of the required steps.

Usage

To create structures of your own objects you'll need a triangular (2D planar) .obj file or a tetrahedral (3D volume) .msh file as input, respectively. Suitable tet-inputs can be generated using fTetWild using the --no-binary option.

Generating a Stress Field

Boundary conditions, i.e., fixed vertices and applied forces, can be defined in a .frc file using the following structure:

• First, the input file file (file.obj/msh) and mesh type type (tri/quad/tet/hex) are specified.
• Keywords fix and flx at the start of a line specify if selected vertices are either fixed or flexible (where forces are applied).
• Vertex selection has two modes d (dimension) and r (radius). Thereby, d is followed by two numbers, the first (int) indicating the dimension (0=x, 1=y, 2=z) and the second (float) to specify the position of the selection plane. This will select all vertices on the positive side of the plane, i.e., above the threshold. For the inverse, precede the dimension with a -. The radius selection is defined by a point coordinate (two floats in 2D, three in 3D) and a radius.
• The forces acting on the flexible vertices are given with the vec keyword as simple 2D or 3D vectors, respectively. Multiple selections can be given different force vectors and are simply assigned by their order in the force-file.

Lets examine the included bar2D.frc as an example:

file bar2D.obj		# the input file
type tri		# input type
fix d -0 -1		# fixed vertex selection
flx r 1 0 0.01		# flex vertex selection A
flx r 0 -0.5 0.01	# flex vertex selection B
vec 1 0			# force acting on A
vec 0 -1		# force acting on B

• The -0 in the fixed vertex selection indicates that we want to select all vertices below a threshold (-) in x-dimension (0), using a threshold value of -1.
• Vertex selection A includes all vertices around a point (1, 0) within a radius of 0.01.
• Vertex selection B includes all vertices around a point (0, -0.5) within a radius of 0.01.
• Force vectors applied on selected vertices A and B are (1, 0) and (0, -1), respectively.

Fixed and flexible vertex selections include only vertices from the objects boundary hull. One can specify multiple force-files for the same input object, i.e., as exemplified with the included buddha. Lines can be commented-out using #. Once the forces are applied by the FemObject.py, the FEM simulation data is stored in a .stress file.

Generating a Hexahedral Lattice

Starting from the stress field just created, we can initialize a cubification object using various parameters, as exemplified in the CubeObject.py file. By default, these settings are stored in .cfg files, thus can be reloaded again. Parameters are self-explanatory and described in the paper. Resulting quad- or hex-meshes are stored in .ply or .mesh files, respectively.

Evaluation

Optimized results from the cubification are still solid objects. To produce actual lightweight designs, they are again loaded as FemObject.py using the same initial force-file but setting the loadResult = True flag. Then we can materialize the object using micro-structures and evaluate the compliance of the new optimized design by reapplying the specified boundary conditions.

Citation

You can cite the paper with:

@Article{bukenberger2024stress,
  author    = {Bukenberger, Dennis R. and Wang, Junpeng and Wu, Jun and Westermann, Rüdiger},
  journal   = {Computer Graphics Forum},
  title     = {{Stress-Aligned Hexahedral Lattice Structures}},
  year      = {2024},
  issn      = {1467-8659},
  pages     = {e15265},
  volume    = {43.7},
  doi       = {10.1111/cgf.15265},
  publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
}